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Gustaf Ahdritz, Tian Qin, Nikhil Vyas, Boaz Barak, Benjamin L. Edelman

We study the feasibility of identifying epistemic uncertainty (reflecting a lack of knowledge), as opposed to aleatoric uncertainty (reflecting entropy in the underlying distribution), in the outputs of large language models (LLMs) over free-form text. In the absence of ground-truth probabilities, we explore a setting where, in order to (approximately) disentangle a given LLM's uncertainty, a significantly larger model stands in as a proxy for the ground truth. We show that small linear probes trained on the embeddings of frozen, pretrained models accurately predict when larger models will be more confident at the token level and that probes trained on one text domain generalize to others. Going further, we propose a fully unsupervised method that achieves non-trivial accuracy on the same task. Taken together, we interpret these results as evidence that LLMs naturally contain internal representations of different types of uncertainty that could potentially be leveraged to devise more informative indicators of model confidence in diverse practical settings.

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Davis Brown, Nikhil Vyas, Yamini Bansal

In this study, we investigate whether the representations learned by neural networks possess a privileged and convergent basis. Specifically, we examine the significance of feature directions represented by individual neurons. First, we establish that arbitrary rotations of neural representations cannot be inverted (unlike linear networks), indicating that they do not exhibit complete rotational invariance. Subsequently, we explore the possibility of multiple bases achieving identical performance. To do this, we compare the bases of networks trained with the same parameters but with varying random initializations. Our study reveals two findings: (1) Even in wide networks such as WideResNets, neural networks do not converge to a unique basis; (2) Basis correlation increases significantly when a few early layers of the network are frozen identically. Furthermore, we analyze Linear Mode Connectivity, which has been studied as a measure of basis correlation. Our findings give evidence that while Linear Mode Connectivity improves with increased network width, this improvement is not due to an increase in basis correlation.

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Nikhil Vyas, Depen Morwani, Rosie Zhao, Gal Kaplun, Sham Kakade, Boaz Barak

The success of SGD in deep learning has been ascribed by prior works to the implicit bias induced by high learning rate or small batch size ("SGD noise"). While prior works that focused on offline learning (i.e., multiple-epoch training), we study the impact of SGD noise on online (i.e., single epoch) learning. Through an extensive empirical analysis of image and language data, we demonstrate that large learning rate and small batch size do not confer any implicit bias advantages in online learning. In contrast to offline learning, the benefits of SGD noise in online learning are strictly computational, facilitating larger or more cost-effective gradient steps. Our work suggests that SGD in the online regime can be construed as taking noisy steps along the "golden path" of the noiseless gradient flow algorithm. We provide evidence to support this hypothesis by conducting experiments that reduce SGD noise during training and by measuring the pointwise functional distance between models trained with varying SGD noise levels, but at equivalent loss values. Our findings challenge the prevailing understanding of SGD and offer novel insights into its role in online learning.

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Nikhil Vyas, Alexander Atanasov, Blake Bordelon, Depen Morwani, Sabarish Sainathan, Cengiz Pehlevan

We study the effect of width on the dynamics of feature-learning neural networks across a variety of architectures and datasets. Early in training, wide neural networks trained on online data have not only identical loss curves but also agree in their point-wise test predictions throughout training. For simple tasks such as CIFAR-5m this holds throughout training for networks of realistic widths. We also show that structural properties of the models, including internal representations, preactivation distributions, edge of stability phenomena, and large learning rate effects are consistent across large widths. This motivates the hypothesis that phenomena seen in realistic models can be captured by infinite-width, feature-learning limits. For harder tasks (such as ImageNet and language modeling), and later training times, finite-width deviations grow systematically. Two distinct effects cause these deviations across widths. First, the network output has initialization-dependent variance scaling inversely with width, which can be removed by ensembling networks. We observe, however, that ensembles of narrower networks perform worse than a single wide network. We call this the bias of narrower width. We conclude with a spectral perspective on the origin of this finite-width bias.

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Nikhil Vyas, Sham Kakade, Boaz Barak

There is a growing concern that learned conditional generative models may output samples that are substantially similar to some copyrighted data $C$ that was in their training set. We give a formal definition of $\textit{near access-freeness (NAF)}$ and prove bounds on the probability that a model satisfying this definition outputs a sample similar to $C$, even if $C$ is included in its training set. Roughly speaking, a generative model $p$ is $\textit{$k$-NAF}$ if for every potentially copyrighted data $C$, the output of $p$ diverges by at most $k$-bits from the output of a model $q$ that $\textit{did not access $C$ at all}$. We also give generative model learning algorithms, which efficiently modify the original generative model learning algorithm in a black box manner, that output generative models with strong bounds on the probability of sampling protected content. Furthermore, we provide promising experiments for both language (transformers) and image (diffusion) generative models, showing minimal degradation in output quality while ensuring strong protections against sampling protected content.

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Nikhil Vyas, Yamini Bansal, Preetum Nakkiran

The ``Neural Tangent Kernel'' (NTK) (Jacot et al 2018), and its empirical variants have been proposed as a proxy to capture certain behaviors of real neural networks. In this work, we study NTKs through the lens of scaling laws, and demonstrate that they fall short of explaining important aspects of neural network generalization. In particular, we demonstrate realistic settings where finite-width neural networks have significantly better data scaling exponents as compared to their corresponding empirical and infinite NTKs at initialization. This reveals a more fundamental difference between the real networks and NTKs, beyond just a few percentage points of test accuracy. Further, we show that even if the empirical NTK is allowed to be pre-trained on a constant number of samples, the kernel scaling does not catch up to the neural network scaling. Finally, we show that the empirical NTK continues to evolve throughout most of the training, in contrast with prior work which suggests that it stabilizes after a few epochs of training. Altogether, our work establishes concrete limitations of the NTK approach in understanding generalization of real networks on natural datasets.

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Mitali Bafna, Jack Murtagh, Nikhil Vyas

We give a new algorithm for approximating the Discrete Fourier transform of an approximately sparse signal that has been corrupted by worst-case $L_0$ noise, namely a bounded number of coordinates of the signal have been corrupted arbitrarily. Our techniques generalize to a wide range of linear transformations that are used in data analysis such as the Discrete Cosine and Sine transforms, the Hadamard transform, and their high-dimensional analogs. We use our algorithm to successfully defend against well known $L_0$ adversaries in the setting of image classification. We give experimental results on the Jacobian-based Saliency Map Attack (JSMA) and the Carlini Wagner (CW) $L_0$ attack on the MNIST and Fashion-MNIST datasets as well as the Adversarial Patch on the ImageNet dataset.

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